Contents

cohomology

# Contents

## Idea

Lie algebra cohomology is the intrinsic notion of cohomology of Lie algebras.

There is a precise sense in which Lie algebras $\mathfrak{g}$ are infinitesimal Lie groups. Lie algebra cohomology is the restriction of the definition of Lie group cohomology to Lie algebras.

In ∞-Lie theory one studies the relation between the two via Lie integration.

Lie algebra cohomology generalizes to nonabelian Lie algebra cohomology and to ∞-Lie algebra cohomology.

## Definition

There are several different but equivalent definitions of the cohomology of a Lie algebra.

### As Ext-group or derived functor

The abelian cohomology of a $k$-Lie algebra $\mathfrak{g}$ with coefficients in the left $\mathfrak{g}$-module $M$ is defined as $H^*_{Lie}(\mathfrak{g},M) = Ext_{U\mathfrak{g}}^*(k,M)$ where $k$ is the ground field understood as a trivial module over the universal enveloping algebra $U\mathfrak{g}$. In particular it is a derived functor.

### Via resolutions

Before this approach was advanced in Cartan-Eilenberg’s Homological algebra, Lie algebra cohomology and homology were defined by Chevalley-Eilenberg with a help of concrete Koszul-type resolution which is in this case a cochain complex

$Hom_{\mathfrak{g}}(U\mathfrak{g}\otimes_k \Lambda^* \mathfrak{g},M)\cong Hom_k(\Lambda^* \mathfrak{g},M),$

where the first argument $U\mathfrak{g}\otimes_k \Lambda^* \mathfrak{g}$ is naturally equipped with a differential to start with (see below).

WHERE BELOW?

The first argument in the Hom, i.e. $U\mathfrak{g}\otimes_k \Lambda^* \mathfrak{g}$ is sometimes called the Chevalley-Eilenberg chain complex (cf. Weibel); the Chevalley-Eilenberg cochain complex is the whole thing, i.e.

$CE(\mathfrak{g},M) := Hom_{\mathfrak{g}}(U\mathfrak{g}\otimes_k \Lambda^* \mathfrak{g},M)\cong Hom_k(\Lambda^* \mathfrak{g},M).$

If $M$ is a trivial module $k$ then $CE(\mathfrak{g}) := Hom_k(\Lambda^* \mathfrak{g},k)$ and if $\mathfrak{g}$ is finite-dimensional this equals $\Lambda^* \mathfrak{g}^*$ with an appropriate differential and the exterior multiplication gives it a dg-algebra structure.

### Via $\infty$-Lie algebras

As discussed at Chevalley-Eilenberg algebra, we may identify Lie algebras $\mathfrak{g}$ as the duals $CE(\mathfrak{g})$ of dg-algebras whose underlying graded algebra is the Grassmann algebra on the vector space $\mathfrak{g}^*$.

Similarly, a dg-algebra $CE(\mathfrak{h})$ whose underlying algebra is free on a graded vector space $\mathfrak{h}$ we may understand as exibiting an ∞-Lie algebra-structure on $\mathfrak{h}$.

Then a morphism $\mathfrak{g} \to \mathfrak{h}$ of these $\infty$-Lie algebras is by definition just a morphism $CE(\mathfrak{g}) \leftarrow CE(\mathfrak{h})$ of dg-algebras. Such a morphis may be thought of as a cocycle in nonabelian Lie algebra cohomology $H(\mathfrak{g}, \mathfrak{h})$.

Specifically, write $b^{n-1} \mathbb{R}$ for the line Lie n-algebra, the $\infty$-Lie algebra given by the fact that $CE(b^{n-1}\mathbb{R})$ has a single generator in degree $n$ and vanishing differential. Then a morphism

$\mu : \mathfrak{g} \to b^{n-1} \mathbb{R}$

is a cocycle in the abelian Lie algebra cohomology $H^n(\mathfrak{g}, \mathbb{R})$. Notice that dually, by definition, this is a morphism of dg-algebras

$CE(\mathfrak{g}) \leftarrow CE(b^{n-1} \mathbb{R}) : \mu \,.$

Since on the right we only have a single closed degree-$n$ generator, such a morphism is precily a closed degree $n$-element

$\mu \in CE(\mathfrak{g}) \,.$

This way we recover the above definition of Lie algebra cohomology (with coefficient in the trivial module) in terms of the cochain complex cohomology of the CE-algebra.

## Properties

The following lemma asserts that for semisimple Lie algebras $\mathfrak{g}$ only the cohomology $\mathfrak{g} \to b^{n-1} \mathbb{R}$ with coefficients in the trivial module is nontrivial.

###### Proposition

For $\mathfrak{g}$ a finite dimensional semisimple Lie algebra over a field of characteristic 0, and for $V$ a non-trivial finite-dimensional irreducible representation, we have

$H^p(\mathfrak{g}, V) = 0 \;\;\; for\;p \gt 0 \,.$

### Van Est isomorphism

The content of a van Est isomorphism is that the canonical comparison map from Lie group cohomology to Lie algebra cohomology (by differentiation) is an isomorphism whenever the Lie group is sufficiently connected.

### Hochschild-Serre spectral sequence

###### Definition

(relative Lie algebra cohomology with coefficents)

Let

1. $(\mathfrak{g}, [-,-])$ be a Lie algebra of finite dimension;

2. $(V, \rho)$ a $\mathfrak{g}$-Lie algebra module of finite dimension;

3. $\mathfrak{h} \hookrightarrow \mathfrak{g}$ a sub-Lie algebra.

Consider the $\mathbb{N}$-graded vector space

$C^\bullet(\mathfrak{g}, \mathfrak{h}, V) \;\coloneqq\; \left( ( \wedge^\bullet (\mathfrak{g}/\mathfrak{h})^\ast ) \otimes V \right)^{\mathfrak{h}}$

consisting of the $\mathfrak{h}$-invariant elements in the tensor product of $V$ with the exterior algebra of the coset $\mathfrak{g}/\mathfrak{h}$.

On this graded vector space, the dual linear maps of the Lie bracket $[-,-]$, extended as a graded derivation to the exterior algebra, and the Lie algebra action $\rho$ define a differential

$d_{CE} \coloneqq \rho^\ast + [-,-]^\ast \,.$

The resulting cochain complex is the Chevalley-Eilenberg complex of $\mathfrak{g}$ relative $\mathfrak{h}$ with coefficients in $V$.

$H^\bullet(\mathfrak{g}, \mathfrak{h}; V) \;\coloneqq\; \left( \left( ( \wedge^\bullet (\mathfrak{g}/\mathfrak{h})^\ast ) \otimes V \right)^{\mathfrak{h}} , d_{CE} \right)$

is the Lie algebra cohomology of $\mathfrak{g}$ relative $\mathfrak{h}$ with coefficients in $V$.

If in def. $\mathfrak{h} = 0$ then the definition reduces to that of ordinary Lie algebra cohomology with coefficients:

$H^\bullet(\mathfrak{g}, 0; V) = H^\bullet(\mathfrak{g}; V) \,.$
###### Definition

(Lie algebra reductive in ambient Lie algebra)

A sub-Lie algebra

$\mathfrak{h} \hookrightarrow \mathfrak{g}$

is called reductive if the adjoint Lie algebra representation of $\mathfrak{h}$ on $\mathfrak{g}$ is reducible.

(Koszul 50, recalled in e.g. Solleveld 02, def. 2.27)

###### Proposition

(invariants in Lie algebra cohomology computed by relative Lie algebra cohomology)

Let

1. $(\mathfrak{g}, [-,-])$ be a Lie algebra of finite dimension;

2. $(V, \rho)$ a $\mathfrak{g}$-Lie algebra module of finite dimension, which is reducible;

3. $\mathfrak{h} \hookrightarrow \mathfrak{g}$ a sub-Lie algebra which is reductive in $\mathfrak{g}$ (Def. ) in that its adjoint representation on $\mathfrak{g}$ is reducible.

4. such that

$\mathfrak{g} = \mathfrak{h} \ltimes \mathfrak{a}$

is a semidirect product Lie algebra (hence $\mathfrak{a}$ a Lie ideal).

Then the invariants in the Lie algebra cohomology of $\mathfrak{a}$ (either with respect to $\mathfrak{h}$ or all of $\mathfrak{g}$) coincide with the relative Lie algebra cohomology (Def. , using the invariant subcomplex!):

$H^\bullet(\mathfrak{a}; V)^{\mathfrak{h}} \;\simeq\; H^\bullet(\mathfrak{g}, \mathfrak{h}; V) \,.$

Proof via the Hochschild-Serre spectral sequence.

## Examples

Every invariant polynomial $\langle - \rangle \in W(\mathfrak{g})$ on a Lie algebra has a transgression to a cocycle on $\mathfrak{g}$. See ∞-Lie algebra cohomology for more.

For instance for $\mathfrak{g}$ a semisimple Lie algebra, there is the Killing form $\langle - ,- \rangle$. The corresponding 3-cocycle is

$\mu = \langle -, [-,-] \rangle : CE(\mathfrak{g}) \,,$

that is: the function that sends three Lie algebra elements $x, y, z$ to the number $\mu(x,y,z) = \langle x, [y,z]\rangle$.

On the super Poincare Lie algebra in dimension (10,1) there is a 4-cocycle

$\mu_4 = \bar \psi \wedge \Gamma^{a b} \Psi\wedge e_a \wedge e_b \in CE(\mathfrak{siso}(10,1))$

## Extensions

Every Lie algebra degree $n$ cocycle $\mu$ (with values in the trivial model) gives rise to an extension

$b^{n-2} \mathbb{R} \to \mathfrak{g}_{\mu} \to \mathfrak{g} \,.$

In the language of ∞-Lie algebras this was observed in (BaezCrans Theorem 55).

In the dual dg-algebra language the extension is lust the relative Sullivan algebra

$CE(\mathfrak{g}_\mu) \leftarrow CE(\mathfrak{g})$

obtained by gluing on a rational $n$-sphere. By this kind of translation between familiar statements in rational homotopy theory dually into the language of ∞-Lie algebras many useful statements in ∞-Lie theory are obtained.

Examples

### Ordinary Lie algebras

Accounts of the standard theory of Lie algebra cohomology include

### Super Lie algebras

The cohomology of super Lie algebras is analyzed via normed division algebras in

This subsumes some of the results in

The cohomology of the super Poincare Lie algebra in low dimensions $\leq 5$ is analyzed in

• Supersymmetry algebra cohomology I: Definition and general structure J. Math. Phys.51:122302, 2010, arXiv

Supersymmetry algebra cohomology II: Primitive elements in 2 and 3 dimensions J. Math. Phys. 51 (2010) 112303 (arXiv)

Supersymmetry algebra cohomology III: Primitive elements in four and five dimensions (arXiv)

and in higher dimensions more generally in

• Michael Movshev, Albert Schwarz, Renjun Xu, Homology of Lie algebra of supersymmetries (arXiv) .

### Extensions

The ∞-Lie algebra extensions $b^{n-2} \to \mathfrak{g}_\mu \to \mathfrak{g}$ induced by a degree $n$-cocycle are considered

• John Baez, Alissa Crans, around theorem 55 in Higher-Dimensional Algebra VI: Lie 2-Algebras, Theory and Applications of Categories 12 (2004), 492-528. arXiv